The external noise paradigm and perceptual template model (PTM) have successfully been applied to characterize observer properties and mechanisms of observer state changes (e.g. attention and perceptual learning) in several research domains, focusing on individual level analysis. In this study, we developed a new hierarchical Bayesian perceptual template model (HBPTM) to model the trial-by-trial data from all individuals and conditions in a published spatial cuing study within a single structure and compared its performance to that of a Bayesian Inference Procedure (BIP), which separately infers the posterior distributions of the model parameters for each individual subject without the hierarchical structure. The HBPTM allowed us to compute the joint posterior distribution of the hyperparameters and parameters at the population, observer, and experiment levels and make statistical inferences at all these levels. In addition, we ran a large simulation study that varied the number of observers and number of trials in each condition and demonstrated the advantage of the HBPTM over the BIP across all the simulated datasets. Although it is developed in the context of spatial attention, the HBPTM and its extensions can be used to model data from the external noise paradigm in other domains and enable predictions of human performance at both the population and individual levels.

*N*, and proportional constant of multiplicative noise

_{a}*N*) and additional observer state-dependent parameters associated with, for example, attention (internal additive noise reduction

_{m}*A*, external noise exclusion

_{a}*A*, and multiplicative internal noise reduction

_{f}*A*). So far, all existing applications of the PTM have focused at the individual level. Typically, the maximum likelihood or least-squares procedure is used to fit variants of the PTM that consists of different numbers of observer state-dependent parameters (e.g. a single

_{m}*A*, a single

_{a}*A*, or both

_{f}*A*and

_{a}*A*) to each individual observer's TvC functions or psychometric functions in multiple external noise conditions, and a nested model comparison procedure is used to identify the best fitting model and therefore the associated mechanism(s) for the individual (Dosher & Lu, 2000b; Lu & Dosher, 1998; Lu & Dosher, 2013). Occasionally, bootstrap procedures are used to estimate the variabilities of the best-fitting model parameters (Lu & Dosher, 2013). Although they provide excellent point estimates of the best fitting PTM parameters at the individual level, and often the results in different individuals are similar, these modeling procedures treat data from each individual separately without explicitly capitalizing on potential regularities of model parameters across individuals. In addition, they are not designed for statistical inference at the populational level.

_{f}*d*′ in condition (

*c*,

*N*,

_{ext}*cue*), where

*c*is the pseudo-character signal contrast,

*N*is the standard deviation of external noise contrast, and

_{ext}*cue*is either pre or simultaneous, is a function of six model parameters (

*N*,

_{a}*N*, β, γ,

_{m}*A*, and

_{a}*A*) and two stimulus parameters (

_{f}*c*and

*N*; Dosher & Lu, 2000b; Lu & Dosher, 1998; Lu & Dosher, 2000):

_{ext}*N*is the standard deviation of the internal additive noise,

_{a}*N*is the proportional constant of the multiplicative noise, β is the gain of the perceptual template, γ is the exponent of the transducer,

_{m}*A*reflects internal additive noise reduction by attention,

_{a}*A*reflects external noise exclusion by attention.

_{f}^{1}

*A*and

_{a}*A*depend on the cuing condition. The probability of obtaining a correct response in a single trial is:

_{f}*G*(.) are the probability density and cumulative probability functions of a standard Gaussian distribution. The probability of obtaining

*M*correct responses from a total of

*T*trials in a single condition is described by a binomial distribution

*B*:

*stimulus enhancement*reduces contrast thresholds in the region of zero or low external noise (see Figures 2a, b), accounting for effects of attention in the absence of external noise. Mathematically equivalent to internal additive noise reduction, it corresponds to claims of perceptual enhancement (Posner, Nissen, & Ogden, 1978).

*External noise exclusion*reduces contrast thresholds in the region of high external noise (see Figures 2c, d), where there is external noise to exclude, by focusing perceptual analysis on the appropriate time, spatial region, and/or content characteristics of the signal stimulus (Dosher & Lu, 2000b; Shiu & Pashler, 1994).

*Multiplicative internal noise reduction*reduces contrast thresholds throughout the entire range of external noise levels (see Figures 2e, f). In addition, measuring TvC functions at two or more criterion performance levels along the psychometric function resolves the individual contribution of each mechanism in when multiple mechanisms are involved (Dosher & Lu, 2000a; Lu & Dosher, 2000). In prior applications of the PTM, only stimulus enhancement and external noise exclusion have been observed, so these two mechanisms of attention are examined in our analysis.

**θ***to denote the PTM parameters for individual*

_{ij}*i*in the

*j*th test, which is the

*j*th repetition of the whole experiment with all the stimulus contrast, external noise, and cuing conditions.

**S**

_{ijk},

*T*, and

_{ijk}*M*denote, respectively, the stimulus parameters (

_{ijk}*c*,

*N*, and cue), the numbers of total trials and correct responses for individual

_{ext}*i*in the

*k*th condition of the

*j*th repetition, where condition

*k*denotes each combination of

*c*,

*N*, and cue. Table 1 shows a model structure with external noise exclusion in central cuing and a combination of external noise exclusion and internal noise reduction in peripheral cuing, which agrees with the previous analysis of the study (Lu & Dosher, 2000). Later, we will consider additional models in which central cuing causes both external noise exclusion and internal noise reduction, and a model in which attention has no effect. The system nonlinearity parameter γ of the PTM is assumed to be equal in central and peripheral cuing, consistent with many applications of the PTM (Dosher & Lu, 2000a; Dosher & Lu, 2000b; Lu & Dosher, 2000). (By convention, the

_{ext}*A*and

_{a}*A*parameters are set to 1 in simultaneous, or unattended, conditions, and in both conditions if the respective attention mechanisms are not effective.

_{f}*A*and

_{a}*A*< 1 if attention improves performance.)

_{f}*M*correct responses in

_{ijk}*T*trials as:

_{ijk}

**θ**_{1: I, 1: J}, is:

*k*runs through all the (

*c*,

*N*, and cuing) combinations for each individual, with

_{ext}*K*= 288 for the three observers who participated in both the central and peripheral cuing experiments, and

_{i}*K*= 144 for the two observers who only participated in the central cuing experiment.

_{i}

**θ***for each of the observers in all the tests:*

_{ij}*are used to characterize distributions of observer properties (template gain, transducer nonlinearity, internal additive and multiplicative noise, and effects of attention) at the population level, hyperparameters*

**η**

**τ**_{i}are used to characterize them for individual observer

*i*, and parameters

**θ**_{i,j}are used to characterize them for individual observer

*i*in the

*j*th test (Table 3). These hyperparameters and parameters are related through conditional probability: parameters

**θ**_{i,j}at the test level are conditioned on the hyperparameters at the individual level, which are in turn conditioned on the hyperparameters

*at the population level:*

**η***p*(

*) is modeled as a mixture of 10-dimensional truncated Gaussian distributions \({\mathcal{N}_t}\) with mean*

**η***, standard deviation*

**µ****σ**(Table 4), which have distributions

*p*(

*) and*

**µ***p*(

**σ**):

*p*(

**τ**_{i}|

*) is modeled as a mixture of 10-dimensional truncated Gaussian distributions \({\mathcal{N}_t}\) with mean*

**η**

**ρ**_{i}and standard deviation

**δ**(see Table 4), which have distributions

*p*(

**ρ**_{i}|

*) and*

**η***p*(

**δ**):

*,*

**µ****σ**, and

**δ**.

_{0,min}and μ

_{0,max}specified in Table 5, and Γ(5000, 3) is a Gamma distribution with a shape parameter of 5000 and a rate parameter of 3. Γ(5000, 3) is used as the prior for all the

**σ**and

**δ**in the model. The zero minima in Tables 4 and 5 ensure that all PTM parameters are positive. The maxima reflect prior knowledge about theoretical boundaries in the PTM (Dosher & Lu, 2000a; Klein & Levi, 2009; Lu & Dosher, 2008).

**μ**, 10

**σ**, 40

**ρ**

_{i}, 10

**δ**, and 40

**θ**

_{i,j}(

**ρ**

_{i}and

**θ**

_{i,j}are 5 dimensional for observers 4 and 5 because they only participated in the central cuing condition). We used the Markov Chain Monte Carlo (MCMC) sampling algorithm in JAGS (Plummer, 2003) to compute the joint posterior distribution of all the hyperparameters and parameters in Equation 11. Three independent MCMC chains were simulated. Each MCMC generated 15,000 kept samples (thinning ratio = 10) via a random walk process after 30,000 burn-in and 100,000 adaptation steps. The same procedure was used to compute the posterior distributions of all the 40

**θ**

_{i,j}′

*s*in Equation 5 in the BIP. Convergence of each parameter was evaluated with Gelman and Rubin's diagnostic rule (Gelman & Rubin, 1992).

*BPIC*= 42867.4 vs 43082.3), and provided an equivalent fit to the data compared to the model with internal additive noise reduction and external noise exclusion in both central and peripheral cuing (

*BPIC*= 42860.5). The results are consistent with the conclusions of the original study (Lu & Dosher, 2000) and a related study of central cuing only (Dosher & Lu, 2000b). We report the results from the main model in subsequent sections.

*Distributions*

**η****η**(

**1**) and

**η**(

**6**), were 0.0241 ± 0.0318 and 0.0264 ± 0.0346 in the endogenous and exogenous cuing experiments; the proportional constants of multiplicative noise,

**η**(

**2**) and

**η**(

**7**), were 0.3122 ± 0.0576 and 0.3292 ± 0.0633; the template gains,

**η**(

**2**) and

**η**(

**7**), were 1.090 ± 0.061 and 1.023 ± 0.069; and the exponent of the transducer function,

**η**(

**4**) was 2.580 ± 0.130. Representing endogenous cuing effects on external noise at the population level,

**η**(

**5**) had a mean of 0.8613 and 95% HWCI of 0.0619, indicating that endogenous cuing significantly excluded external noise. Representing exogenous cuing effects on external noise at the population level,

**η**(

**9**) had a mean of 0.8317 and 95% HWCI of 0.0695, indicating that exogenous cuing also significantly excluded external noise. Representing exogenous cuing effects on internal additive noise at the population level,

**η**(

**10**) had a mean of 0.7627 and HWCI of 0.1130, indicating that exogenous cuing significantly reduced internal additive noise.

**τ**Distributions

**τ**

_{i}(

**5**) ranged from 0.8483 to 0.8726, with 95% HWCI between 0.0674 and 0.0685, indicating that endogenous cuing significantly excluded external noise across all observers. The average

**τ**

_{i}(

**5**) across the five observers was 0.8614, with a 95% HWCI of 0.0388. In comparison, the coefficient of external noise exclusion in Lu and Dosher (2000) ranged from 0.8190 to 0.8872. Representing exogenous cuing effects on external noise at the observer level,

**τ**

_{i}(

**9**) ranged from 0.8268 to 0.8359, with 95% HWCI between 0.0709 and 0.0735, indicating that exogenous cuing also significantly excluded external noise across all observers. The average

**τ**

_{i}(

**9**) across the three observers was 0.8316, with a 95% HWCI of 0.0508. Representing exogenous cuing effects on internal additive noise at the observer level,

**τ**

_{i}(

**10**) ranged from 0.7606 to 0.7666, with 95% HWCI between 0.1151 and 0.1168, indicating that exogenous cuing significantly reduced internal additive noise across all observers. The average

**τ**

_{i}(

**10**) across the three observers was 0.7627, with a 95% HWCI of 0.1020. In comparison, the coefficient of external noise exclusion ranged from 0.8049 to 0.8348 and the coefficient of internal noise reduction ranged from 0.7628 to 0.8575 in Lu and Dosher (2000).

**θ**Distributions

**θ**

_{11}(

**m**) for observer 1. Table 8 lists correlation coefficients between pairs of

**θ**

_{11}(

*). Large negative correlations were found between*

**m****θ**

_{11}(

**1**) and

**θ**

_{11}(

**4**) (−0.9101), and

**θ**

_{11}(

**6**) and

**θ**

_{11}(

**4**) (−0.8834), reflecting tradeoffs between the magnitudes of internal additive noise and the exponent of the transducer function in central and peripheral cuing, respectively. The large positive correlations between

**θ**

_{11}(

**1**) and

**θ**

_{11}(

**6**) (0.8166) reflected correlations of the magnitudes of internal additive noise in central and peripheral cueing. Large positive correlations between

**θ**

_{11}(

**3**) and

**θ**

_{11}(

**5**) (0.5027), and

**θ**

_{11}(

**8**) and

**θ**

_{11}(

**9**) (0.5656) reflected correlations between template gain and external noise exclusion. Similar results were found for the other observers (Supplementary Materials).

**θ**

_{ij}(

*) from the HBPTM and BIP (i.e. the estimates of the observer level parameters). Averaged across all the observers and parameters, the ratio of the mean of the posterior distributions was 0.96 ± 0.27 (mean ± SD), indicating that the expected values of the posterior distributions of the parameters from the two methods were essentially equivalent. Averaged across all the observers and parameters, the ratio of the 95% HWCI of the posterior distributions was 0.68 ± 0.22 (mean ± SD), indicating that the 95% HWCI from the HBPTM was about 32% narrower than that from the BIP. We systematically compared the HBPTM and BIP solutions in the simulations described next.*

**m****θ**

_{ij}(

*) from the HBPTM and BIP in each of the 15 sample sizes. Averaged across all the observers and parameters, the ratio of the mean of the posterior distributions was 1.001 ± 0.023 (mean ± SD), indicating that the expected values of the posterior distributions of the parameters from the two methods were equivalent. However, the 95% HWCI of the posterior distributions from the BIP and HBPTM were quite different.*

**m****θ**

_{ij}(

*) from the HBPTM and BIP in different numbers of trials per experimental condition are shown as functions of the number of simulated observers in log10 units in Figure 7. Each 95% HWCI was normalized by the average 95% HWCI of the posterior distribution of the HBPTM solution in the 72 simulated observers and 40 trials per experimental condition dataset. As expected, the average 95% HWCI from the BIP exhibited very little variability as a function of the number of simulated observers since each observer was modeled separately, but decreased with the number of trials per experimental condition across all the parameters. On the other hand, the average 95% HWCI from the HBPTM decreased with both the number of simulated observers and the number of trials per experimental condition across all the parameters. Across different numbers of trials per experimental condition, the average 95% HWCI approached its asymptotic level between about 10 to 40 simulated observers across the different PTM parameters, suggesting that the HBPTM could benefit from increasing the number of observers up to about 40. Interestingly, the spatial attention parameters,*

**m***for central cuing, and*

**A**_{f}*and*

**A**_{f}*for peripheral cuing, reached their asymptotic levels with 20 observers.*

**A**_{a}**θ**

_{ij}(

*) from the HBPTM and BIP in different numbers of trials per experimental condition as functions of the number of simulated observers in log10 units. The early zig in the blue curves in Figures 8b, g was due to the variability of the 95% HWCI from the BIP. First, all the ratios were less than zero, indicating that the average 95% HWCI from the HBPTM was always less than that from the BIP across all the parameters and simulated sample sizes. Second, across all the parameters and number of simulated observers, the reduction of the 95% HWCI was 0.3453 log10 units (or 54.7%) when the number of trials per experimental condition per observer was 10, 0.2696 log10 units (or 46.3%) when the number of trials per experimental condition per observer was 20, and 0.2126 log10 units (or 38.7%) when the number of trials per experimental condition per observer was 40, indicating increased benefits of the HBPTM when the number of trials per experimental condition is smaller. Finally, across all the parameters and numbers of trials per experimental condition per observer, the benefit of HBPTM reached its asymptotic level when the number of observers was about 40.*

**m****θ**

_{ij}(

*) from the HBPTM and BIP were essentially the same, the 95% HWCI from the HBPTM was less than that from the BIP in the analyses of the experimental data. This pattern was also true in the simulation results, where the benefit of the HBPTM in reducing the HWCIs saturated when the number of observers was about 40.*

**m****σ**and

**δ**with covariances (Zhao, Lesmes, Dorr, & Lu, 2021; Zhao, Lesmes, Hou, & Lu, 2021).

**Conflicts of interest/competing interests:**Z.-L.L. holds intellectual property interests in visual function measurement and rehabilitation technologies, and equity interests in Adaptive Sensory Technology, Inc. (San Diego, CA, USA) and Jiangsu Juehua Medical Technology, Ltd (Jiangsu, China). B.A.D. declares no competing interests.

**Statement about availability of data and code**: The data and R-code are available by request.

*Journal of Neuroscience, Psychology, and Economics*, 4(2), 95–110. [CrossRef] [PubMed]

*Biometrika*, 94(2), 443–458. [CrossRef]

*American Journal of Mathematical and Management Sciences*, 31(1-2), 13–38. [CrossRef]

*Journal of Vision*, 21(3), 1. [CrossRef] [PubMed]

*Journal of Vision*, 17(7), 37. [CrossRef] [PubMed]

*Josa*, 46(8), 634–639.

*The Journal of Physiology*, 136(3), 469. [PubMed]

*Perception & Psychophysics*, 28(3), 241–248. [PubMed]

*Proceedings of the National Academy of Sciences*, 111(47), 16961–16966.

*Perception and Psychophysics*, 55(2), 162–179. [PubMed]

*Vision Research*, 61, 144–156. [PubMed]

*Frontiers in Psychology*, 4, 66. [PubMed]

*Vision Research*, 51(13), 1484–1525. [PubMed]

*The Journal of Physiology*, 219(2), 355. [PubMed]

*Journal of Vision*, 15(10), 9. [PubMed]

*Dual-channel portable amblyopia treatment system with perceptual template model*.

*Paper presented at the 2011 4th International Conference on Biomedical Engineering and Informatics (BMEI)*. Shanghai, China, 2011, pp. 1305–1307.

*The Quarterly Journal of Experimental Psychology A: Human Experimental Psychology*, 47(3), 699–739.

*PLoS One*, 9(3), e90579. [PubMed]

*Journal of Vision*, 11(11), 1271.

*Vision Research*, 45(11), 1399–1412. [PubMed]

*Ciba Found Symp*, 163, 165–175; discussion 175-180. [PubMed]

*Journal of Vision*, 6(7), 6.

*Vision Research*, 99, 37–45. [PubMed]

*Iscience*, 25(1), 103683. [PubMed]

*Vision Research*, 44(12), 1257–1271. [PubMed]

*Journal of Vision*, 5(8), 912.

*Proceedings of the National Academy of Sciences*, 95, 13988–13993.

*Vision Research*, 39, 3197–3221. [PubMed]

*Vision Research*, 40, 1269–1292. [PubMed]

*Psychological Science*, 11, 139–146. [PubMed]

*Proceedings of the National Academy of Sciences*, 102(14), 5286–5290.

*Human Information Processing: Vision, Memory, Attention*(pp. 140–165). Washington, DC.: American Psychological Association.

*Journal of Experimental Psychology. Human Perception and Performance*, 14, 188–202. [PubMed]

*Journal of Experimental Psychology: General*, 113, 501–517. [PubMed]

*Psychological Science*, 8(2), 135–139.

*Statistical Science*, 7(4), 457–472.

*Cognitive Science*, 28(2), 167–207.

*Psychological Review*, 120(3), 472. [PubMed]

*Optica Acta: International Journal of Optics*, 17(7), 515–526.

*Journal of Vision*, 13(8), 4. [PubMed]

*Psychological Science*, 14(6), 598–604. [PubMed]

*Science*, 180(4091), 1194–1197. [PubMed]

*Investigative Ophthalmology & Visual Science*, 61(5), 60. [PubMed]

*Journal of Experimental Psychology. Human Perception and Performance*, 22, 780–787. [PubMed]

*Perception & Psychophysics*, 53, 221–230. [PubMed]

*Journal of the Optical Society of America. A, Optics, Image Science, and Vision*, 71(5), 574–581.

*Attention, Perception, & Psychophysics*, 76(8), 2286–2304. [PubMed]

*Journal of Vision*, 14(13), 9. [PubMed]

*Frontiers in Neuroscience*, 15, 657.

*Investigative Ophthalmology & Visual Science*, 57(12), 1512.

*Journal of Vision*, 9(11), 24.

*Neurobiology of Attention*. New York, NY: Elsevier.

*Journal of the Optical Society of America. A, Optics, Image Science, and Vision*, 26(11), B43–B58. [PubMed]

*Journal of Vision*, 12(9), 1366.

*Frontiers in Psychology*, 5, 977. [PubMed]

*Neural Computation*, 26(11), 2465–2492. [PubMed]

*Journal of the Optical Society of America. A, Optics, Image Science, and Vision*, 26(11), B110–B126. [PubMed]

*Doing Bayesian data analysis: A tutorial with R, JAGS, and Stan*. San Diego, CA: Academic Press.

*Cognitive Science*, 30(3), 1–26. [PubMed]

*Journal of Mathematical Psychology*, 55(1), 1–7.

*Journal of the Optical Society of America. A, Optics, Image Science, and Vision*, 20(7), 1434–1448. [PubMed]

*Vision Research*, 46, 3160–3176. [PubMed]

*Journal of Vision*, 6(6), 519.

*Vision Research*, 49(10), 1194–1204. [PubMed]

*Vision Research*, 46(15), 2315–2327. [PubMed]

*Proceedings of the National Academy of Sciences*, 102(15), 5624–5629.

*Vision Research*, 45(19), 2500–2510. [PubMed]

*Vision Research*, 38, 1183–1198. [PubMed]

*Journal of the Optical Society of America. A, Optics, Image Science, and Vision*, 16, 764–778. [PubMed]

*Journal of Experimental Psychology: Human Perception and Performance*, 26(5), 1534. [PubMed]

*Journal of the Optical Society of America. A, Optics, Image Science, and Vision*, 18(9), 2041–2053. [PubMed]

*Journal of Vision*, 4(1), 5.

*Journal of Vision*, 4(10), 10.

*Neurobiology of Attention*(pp. 448–453): New York, NY: Elsevier.

*Psychological Review*, 115(1), 44–82. [PubMed]

*Learning & Perception*, 1(1), 19–36. [PubMed]

*Visual psychophysics: From laboratory to theory*. Cambridge, MA: MIT Press.

*Vision Research*, 44(12), 1333–1350. [PubMed]

*Journal of Vision*, 2(4), 4.

*Vision Research*, 49(10), 1081–1096. [PubMed]

*Investigative Ophthalmology & Visual Science*, 63(7), 2328–2328.

*Cognitive Psychology*, 63(2), 61–92. [PubMed]

*Cognition*, 217, 104888. [PubMed]

*Perception & Psychophysics*, 69(6), 1009–1021. [PubMed]

*Vision Research*, 45(21), 2759–2772. [PubMed]

*Attention and performance 14: Synergies in experimental psychology, artificial intelligence, and cognitive neuroscience*(pp. 219–243). Cambridge, MA: The MIT Press.

*Journal of Mathematical Psychology*, 55(1), 57–67.

*Computational Brain & Behavior*, 1(2), 184–213.

*Journal of Cognitive Neuroscience*, 31(12), 1976–1996. [PubMed]

*Journal of Vision*, 11(11), 1254.

*Attention and Performance XI*(pp. 205–220). Hillsdale, NJ: Erlbaum.

*Investigative Ophthalmology & Visual Science*, 61(7), 4269.

*Journal of Experimental Psychology: Human Perception and Performance*, 19, 108–130. [PubMed]

*Vision Research*, 31(7-8), 1399–1415. [PubMed]

*Journal of Vision*, 17(10), 1110.

*Scientific Reports*, 7(1), 1–12. [PubMed]

*Journal of the Optical Society of America. A, Optics, Image Science, and Vision*, 4(12), 2355–2365.

*Effects of visual noise*. (PhD dissertation). Cambridge, UK: University of Cambridge,

*Vision: Coding and Efficiency*, pp. 3–24. Cambridge, MA: Cambridge University Press.

*JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling*.

*Paper presented at the Proceedings of the 3rd international workshop on distributed statistical computing*. Retrieved from https://www.R-project.org/conferences/DSC-2003/.

*Journal of the Optical Society of America. A, Optics, Image Science, and Vision*, 60(6), 842–848.

*Chronometric explorations of mind*. Oxford, England: Lawrence Erlbaum.

*Quarterly Journal of Experimental Psychology*, 32(1), 3–25. [PubMed]

*Modes of perceiving and processing information*(pp. 137–157). Hillsdale, NJ: Erlbaum.

*Journal of Neurophysiology*, 110(6), 1346–1356. [PubMed]

*Psychonomic Bulletin & Review*, 12(4), 573–604. [PubMed]

*Psychometrika*, 68(4), 589–606.

*R: A language and environment for statistical computing, Version 2.9. 2*. Vienna, Austria: R Foundation for Statistical Computing. Retrieved from https://cran.-archive.r-project.org/bin/windows/base/old/2.9.2/.

*Stevens' Handbook of Experimental Psychology, Vol. 1: Perception and Motivation; Vol. 2: Learning and Cognition*(2nd ed.) (pp. 739–811). Oxford, England: John Wiley & Sons.

*Journal of Experimental Psychology: Learning, Memory, & Cognition*, 14(3), 562–569.

*Journal of Experimental Psychology. Human Perception and Performance*, 20, 1037–1054.

*Psychol Rev*, 116(2), 283–317. [PubMed]

*Vision Research*, 44, 1297–1320. [PubMed]

*Covert attention does NOT affect contrast sensitivity*.

*Paper presented at the Vision Sciences Society Annual Meeting Abstract (2nd annual meeting)*, Sarasota, Florida. Journal of Vision, 2, 436.

*Journal of the Optical Society of America. A, Optics, Image Science, and Vision*, 34(6), 870–880. [PubMed]

*Handbook of perception and performance*(Vol. 1, pp. 1–85). Hoboken, NJ: Wiley & Sons.

*Journal of the Optical Society of America. A, Optics, Image Science, and Vision*, 62(10), 1221–1232.

*Investigative Ophthalmology & Visual Science*, 43(13), 2916. [PubMed]

*Investigative Ophthalmology & Visual Science*, 51(4), 2294–2299. [PubMed]

*Journal of Neuroscience*, 31(17), 6535–6541. [PubMed]

*Cephalalgia*, 32(2), 125–139. [PubMed]

*Vision Research*, 152, 3–9. [PubMed]

*Vision Research*, 46, 3748–3760. [PubMed]

*Journal of Vision*, 15(10), 12. [PubMed]

*Frontiers in Neuroscience*, 16, 873671. [PubMed]

*Frontiers in Psychology*, 12, 740759. [PubMed]

*Translational Vision Science & Technology*, 10(12), 18. [PubMed]

*Journal of Vision*, 21(12), 9. [PubMed]